14.11: Finding Scholarly Articles
- Examine ways to find scholarly articles and books using library databases and catalogs
Finding Books
Scholarly articles are often found in journals, which compile several peer-reviewed articles on similar topics in the same place. You will often find these articles in online databases or in the periodicals section of your library, but it is still valuable to find pertinent information in actual, physical, books. Books cover virtually any topic, fact or fiction. For research purposes, you will probably be looking for books that synthesize all the information on one topic to support a particular argument or thesis. They will be especially beneficial if you want lots of information on a topic or want to put your topic in context with other important issues.
To find books, you should look in the library catalog, which is typically the main search bar located on the library homepage. The catalog includes books, reference books, media, maps, and titles of periodicals (like magazines, journals, and newspapers). Note that the catalog does not search for articles within periodicals and journals, and you’ll need to utilize a separate article search to find those.
You can locate the book by finding the call number. Call numbers are arranged in alphanumeric order. The call number is based on the book’s subject, author’s last name, and publication date. Call numbers are designated based on the library’s classification system, which determines how books are organized. Many academic libraries use the Library of Congress Classification, while others use the Dewey Decimal System.
Let’s say you are looking for the book called Cyberspace romance: the psychology of online relationships by Monica Whitty and Adrian Carr. You searched in the library catalog and found the call number: HQ 801.82 .W55 2006. Here’s what that means:
- HQ: Subclass HQ refers to The Family, Marriage, Women
- 801.82 refers to Man-Woman relationships, Courtship, Dating
- W refers to the first author’s last name, “Whitty”
- 2006 is the year the book was published
Finding Articles in Databases
So far you have learned how to locate a book you want on the library shelves. What if your project also requires scholarly articles?
To find scholarly articles, you need to look in a database. A research database lets you search across the text of millions of articles published in thousands of academic journals. General databases have a little bit of everything (like a big retail store). Examples of general databases include Google Scholar, the library articles search, or JSTOR. These are good starting points when you’re starting out and shopping around for articles on a wide range of topics, but you may find there are too many search results to sift through. If you’re getting too much irrelevant stuff, try a specialized database.
A specialized database—often called a research or library database—allows targeted searching on one or more specific subject areas (i.e., engineering, medicine, Latin American history, etc.), for a specific format (i.e., books, articles, conference proceedings, video, images), or for a specific date range during which the information was published. Most of what specialized databases contain can not be found by Google or Bing.
You can view the transcript for “What’s a Library Database?” here (opens in new window) .
Your library will probably have an “article search” or “database search” link to begin your search. When you search article databases, your results list contains citations to a variety of information sources. Depending on the database, you might also find citations to book chapters or to books. Below is an excerpt of search results in PsycINFO . Note the different types of information sources that appear.
Everything you need to locate your article is in the citation: the title of the article, the author, the title of the journal, the volume and issue number, the date of publication, and page numbers.
You can typically click on the links below the citation to view the text of the article, or your library may direct you to the location of the article.
Finding Sources from Sources
Every source contains rich clues to other useful sources. It’s a treasure map that can lead you to sources you would never find by pure searching. This skill can help you discern a conversation occurring among a set of scholars or writers about your topic. Think of each good source as giving clues along two axes:
- Forward and backward in time. If you look at a source and see in its bibliography that there are fifty references, you can do a quick scan of the titles and authors to look for other sources you might investigate. These previously cited sources give you a rough map of how the topic has been researched to that point. Similarly, you can look at the “cited by” feature within a database (or Google Scholar) to look for other sources who are continuing the conversation and cited your source.
- Side to side across the scholarly conversation. When looking at a source you like, collect key terms, phrases, and names to find other sources that are similar. These other keywords can lead to other types of evidence and examples that offer more coverage of your topic.
Using Keywords and Similar Subjects
If you’re reading a scholarly article in a library database, you can make use of both the keywords (selected by the author) and the subject-terms (usually determined by the database).
If you’re reading a book, you have two options. First, using the book’s call number (generally found on the side or spine of the book), find the book in the stacks. Nearby books should be on a similar subject. You can also go back to the book’s record in the library catalog. Each book is assigned at least one library subject. Click the subject to find other books with the same subject.
Read the Bibliography
When you have finished the article, you can give the Works Cited page a once-over in order to identify any interesting readings that look useful.
Check out this tutorial from Hunter College Libraries to learn how to read the information in a bibliography or works cited page (look in the left column of the screen for instructions).
Search by Author
Academic writers often write on the same topic and publish several books or articles about the topic. Put the author’s name into a database or Google Scholar search and see what else s/he has published about the topic. The authors may have even published an update to the current study you are reading.
Take a look at Marvin’s success in finding information within a specialized database.
O-Prof: Let’s go back to your initial Google search for a minute. Did any Wikipedia articles come up for bottled water?
Marvin: Yeah, and I took a quick look at one of them. But some of my professors say I shouldn’t use Wikipedia.
O-Prof: That’s because the quality of information in Wikipedia varies. It’s monitored by volunteer writers and editors rather than experts, so you should double-check information you find in Wikipedia with other sources. But Wikipedia articles are often good places to get background info and good places to connect with more reliable sources. Did anything in the Wikipedia article seem useful for finding sources on bottled water?
Marvin clicks back to the Wikipedia site.
Marvin: It does mention that the National Resources Defense Council and the Drinking Water Research Foundation have done some studies on the health effects of bottled water (“Bottled Water”).
O-Prof: So, you could go to the websites for these organizations to find out more about the studies. They might even have links to the full reports of these studies, as well as other resources on your topic. Who else might have something to say about the healthfulness of bottled and tap water?
Marvin: Maybe doctors and other health professionals? But I don’t know any I could ask.
O-Prof: You can look in the library’s subject guides or ask the librarian about databases for health professionals . The Cumulative Index to Nursing & Allied Health Literature (CINAHL) database is a good one. Are you logged in to the library? Can you try that one?
Marvin logs in, finds the database, and types in “bottled water AND health.”
Marvin: Here’s an article called “Health Risks and Benefits of Bottled Water.” It’s in the journal Primary Care Clinical Office Practice (Napier and Kodner).
O-Prof: If that’s a peer-reviewed journal, it might be a good source for your paper.
Marvin: Here’s another one: “Socio-Demographic Features and Fluoride Technologies Contributing to Higher Fluorosis Scores in Permanent Teeth of Canadian Children” (Maupome et al.). That one sounds pretty technical.
O-Prof: And pretty narrow, too. When you start using sources written by experts, you move beyond the huge porch of public discourse, where everyone talks about all questions on a general level, into some smaller conversational parlors, where groups of specialists talk about more narrow questions in greater depth. You generally find more detailed and trustworthy knowledge in these smaller parlors. But sometimes the conversation may be too narrow for your needs and difficult to understand because it’s experts talking to experts.
Way ahead of the professor, Marvin’s already started reading about the health risks and benefits of bottled water.
Marvin: Here’s something confusing. The summary of this article on risks and benefits of bottled water says tap water is fine if you’re in a location where there’s good water. Then it says that you should use bottled water if the purity of your water source is in question. So which is better, tap or bottled?
O-Prof: As you read more sources, you begin to realize there’s not always a simple answer to questions. As the CINAHL article points out, the answer depends on whether your tap water is pure enough to drink. Not everyone agrees on the answers, either. When you’re advising your future clients (or in this case, writing your paper), you’ll need to “listen” to what different people who talk about the healthfulness of bottled and tap water have to say. Then you’ll be equipped to make your own recommendation.
Things to Consider
One last tip for your research is to keep an open mind. If you are not finding good sources, don’t get discouraged. Try a different combination of keywords, synonyms, or ask your librarian or professor for help. Keep in mind that you don’t need a perfect source that aligns with your paper. You can take small bits of information from multiple sources and combine them into your own argument.
This video reminds us that there is not one perfect source for your research, so you may need to use multiple strategies to find the information you need.
You can view the transcript for “One Perfect Source?” here (opens in new window) .
Contributors and Attributions
- Revision, Adaptation, and Original Content. Provided by : Lumen Learning. License : CC BY: Attribution
- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Writing 101: Research 101 tutorial. Provided by : Duke University. Located at : guides.library.duke.edu/writing101/research101. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Google Scholar image. Provided by : COM Library. Located at : libguides.com.edu/content.php?pid=170634&sid=2517762. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Google Scholar. Provided by : The University of Rhode Island. Located at : http://uri.libguides.com/google/gschol . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Screenshot of library catalog search results. Provided by : Duke University Libraries. Located at : guides.library.duke.edu/writing101/research101. Project : Finding Books. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Use what you have to find more, image of axis. Provided by : Gould Library . Located at : http://gouldguides.carleton.edu/c.php?g=147129&p=963917 . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- One Perfect Source?. Authored by : libncsu. Provided by : North Carolina State University. Located at : https://youtu.be/X2VR5adTjeM . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Walk, Talk, Cook, Eat: A Guide to Using Sources. Authored by : Cynthia R. Haller. Located at : http://www.saylor.org/site/wp-content/uploads/2013/01/writing-spaces-readings-on-writing-vol-2.pdf . Project : Writing Spaces: Readings on Writing Vol. 2. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Refine Your Search. Authored by : Duke University Libraries. Located at : guides.library.duke.edu/writing101/research101. Project : Library 101. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Using Good Sources to Find More Good Sources. Authored by : Wendy Hayden and Stephanie Margolin. Provided by : Hunter College Libraries. Located at : http://library.hunter.cuny.edu/research-toolkit/how-do-i-find-sources/using-good-sources . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- What is a Library Database?. Authored by : What's a library database? . Provided by : RMIT University. Located at : https://www.youtube.com/watch?v=KKIbnNLCh8g&feature=emb_logo . License : Other . License Terms : Standard YouTube License
- Types of Information Sources, books. Authored by : Virginia Tech Libraries. Located at : www.lib.vt.edu/help/research/info-sources.html. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike


27.2B: Cleavage of the Zygote
The process of cleavage is the step of embryogenesis where the zygote divides to produce a cluster of cells known as the morula.
- Describe cleavage of the zygote
Key Points
- Following fertilization a series of rapid cell divisions occur that decrease the cells’ size with each subsequent division—this eventually produces a morula. The different cells derived from cleavage, up to the blastula stage, are called blastomeres.
- For species such as humans, there is little yolk in eggs, and the divisions are relatively symmetrical, or holoblastic.
- For other species, such as birds and reptiles, the presence of yolk dictates uneven meroblastic divisions that produce cells of uneven size and distribution.
- The end of cleavage is known as the midblastula transition and coincides with the onset of zygotic transcription.
- The cells of the morula are at first closely aggregated, but quickly become arranged into an outer or peripheral layer, the trophoblast, which does not contribute to the formation of the embryo proper, and an inner cell mass from which the embryo develops.
Key Terms
- cleavage : In embryology, this is the division of cells in the early embryo.
- trophoblast : The membrane of cells that forms the wall of a blastocyst during early pregnancy and provides nutrients to the embryo, and later develops into part of the placenta.
- zygote : A fertilized egg cell.
- morula : A spherical mass of blastomeres that forms following the splitting of a zygote; it becomes the blastula.
Cell division with no significant growth that produces a cluster of cells that is the same size as the original zygote, is called cleavage. At least four initial cell divisions occur, resulting in a dense ball of at least sixteen cells called the morula.
The different cells derived from cleavage up to the blastula stage are called blastomeres. Depending mostly on the amount of yolk in the egg, the cleavage can be holoblastic (total) or meroblastic (partial).
Cell cleavage : Early development is characterized by cleavage of the zygote, which refers to cell divisions that are not associated with significant growth of the embryo.
Holoblastic cleavage occurs in animals with little yolk in their eggs. These species, such as humans and other mammals, receive nourishment as embryos from the mother via the placenta or milk after birth.
On the other hand, meroblastic cleavage occurs in animals whose eggs have more yolk, such as birds and oviparous reptiles (although some viviparous reptiles also exist). Since cleavage is impeded by the vegetal pole, there is a very uneven distribution and size of cells. Cells are more numerous and smaller at the animal pole of the zygote than at the vegetal pole.
In holoblastic eggs, the first cleavage always occurs along the vegetal–animal axis of the egg, and the second cleavage is perpendicular to the first. From here, the spatial arrangement of blastomeres can follow various patterns, due to different planes of cleavage in various organisms.The end of cleavage is known as the midblastula transition and coincides with the onset of zygotic transcription.
In amniotes, the cells of the morula are at first closely aggregated. However, they quickly become arranged into an outer or peripheral layer, the trophoblast, and an inner cell mass. The trophoblast does not contribute to the formation of the embryo proper; the embryo develops from the inner cell mass.
Fluid collects between the trophoblast and the greater part of the inner cell mass, and thus the morula, is converted into the blastodermic vesicle (also called the blastocyst or blastula). The inner cell mass remains in contact with the trophoblast at one pole of the ovum. This is named the embryonic pole, since it indicates the location where the future embryo will develop.
In the case of monozygotic twins (derived from one zygote), a zygote divides into two separate cells (embryos) at the first cleavage division. Monozygotic twins can also develop from two inner cell masses.
A rare occurrence is the division of a single inner cells mass giving rise to twins. However, if one inner cell mass divides incompletely, the result is conjoined twins. Dizygotic twins is the development of two embryos from two different zygotes.


13.9: Community Intervention Models
The social-ecological model identifies several potential areas for intervention and health promotion within a community at each different circle. Yet, just as individuals have varying levels of readiness to change behaviors, so do communities have varying levels of readiness for intervention (McKenzie et al., 2017). Due to group dynamics, physical environments, and political powers which are involved in every community, we cannot necessarily apply the same behavior change theories to a group that we might to an individual. Several researchers have described different approaches to community interventions, some which emphasize looking for deficits (needs-based) and others that emphasize identifying what is already working well (strengths-based). These different models work via social action, advocacy, capacity building, grassroots organizing, and leadership development in order to cause positive change (Seabert et al., 2021).
Community-based health promotion programs often take a holistic approach, in that they are not limited to a single location (like a hospital), they often have multiple, coordinated interventions, and may also focus on environmental and policy changes to support the desired behaviors (Merzel & D’Afflitti, 2003). There are four major models that most community-based interventions fall into according to McLeroy et al. (2003). Most successful community interventions are not only multi-faceted, they also utilize more than one of these models.
- Community as a setting . Sometimes the community is simply the geographical location in which the intervention takes place. Health promotion programming might take place at various locations within the community, such as city parks, community centers, churches or other faith-based organizations, hospitals, schools, worksites, or other non-profit organizations. Mass media (like billboards, paper flyers, radio, T.V., or Youtube advertisements, etc.) may be used to convey specific health messaging. When the community is the setting, the goal of the intervention is to elicit personal behavioral change. Therefore, any improvement in population health is the sum of all of the individual behaviors. This model may fail to address environmental, economic, or political influences of behavior and health.
- Community as a target . Interventions that target the community seek to make changes to the environment, programs, and/or policies available to community members. Instead of focusing on individual health behaviors as outcomes, these interventions might focus on other environmental or policy measures related to health, such as acreage of park-space per person, or the number of seniors participating in a free lunch program. The assumption is that if the community environment is changed, this will improve healthy behaviors and quality of life.
- Community as a resource . This approach seeks to utilize the people and resources within the community to either drive or at least contribute to health-promoting changes. A public health practitioner might bring a health priority to the community and seek input on intervention design as well as leveraging and supporting current resources, or they might ask the community members themselves to identify the highest priority for the community, and build an intervention from there.
- Community as an agent . This model assumes that the community already has the solutions and resources to its own health problems. It attempts to “meet people where they are” in supporting the community itself to identify needs, come up with solutions, and implement them. This model might fail to address disparities in social determinants of health that many communities face such as economics, environment, and policy.
Public health interventions are indelibly tied to current cultural perceptions and beliefs about society, and their success depends upon how well they are designed, received, and implemented. To what extent does the responsibility for health lie within each person (individualism) or within society (collectivism)? What role should the government play in protecting and promoting health? As McLeroy et al. (2003) states:
Public health is more than a body of theory and intervention methods. We cannot separate how we do public health from why we do public health. Whether we talk about changing behavior, changing community structures, or building community capacity, these changes cannot be separated from our ideals about what constitutes a good community or a good society.


6.1: Voltage Divider Circuits
Let’s analyze a simple series circuit, determining the voltage drops across individual resistors:
Determine the Total Circuit Resistance
From the given values of individual resistances, we can determine a total circuit resistance, knowing that resistances add in series:
Use Ohm’s Law to Calculate Electron Flow
From here, we can use Ohm’s Law (I=E/R) to determine the total current, which we know will be the same as each resistor current, currents being equal in all parts of a series circuit:
Now, knowing that the circuit current is 2 mA, we can use Ohm’s Law (E=IR) to calculate voltage across each resistor:
It should be apparent that the voltage drop across each resistor is proportional to its resistance, given that the current is the same through all resistors. Notice how the voltage across R 2 is double that of the voltage across R 1 , just as the resistance of R 2 is double that of R 1 .
If we were to change the total voltage, we would find this proportionality of voltage drops remains constant:
The voltage across R 2 is still exactly twice that of R 1 ‘s drop, despite the fact that the source voltage has changed. The proportionality of voltage drops (ratio of one to another) is strictly a function of resistance values.
With a little more observation, it becomes apparent that the voltage drop across each resistor is also a fixed proportion of the supply voltage. The voltage across R 1 , for example, was 10 volts when the battery supply was 45 volts. When the battery voltage was increased to 180 volts (4 times as much), the voltage drop across R 1 also increased by a factor of 4 (from 10 to 40 volts). The ratio between R 1 ‘s voltage drop and total voltage, however, did not change:
Likewise, none of the other voltage drop ratios changed with the increased supply voltage either:
Voltage Divider Formula
For this reason a series circuit is often called a voltage divider for its ability to proportion—or divide—the total voltage into fractional portions of constant ratio. With a little bit of algebra, we can derive a formula for determining series resistor voltage drop given nothing more than total voltage, individual resistance, and total resistance:
The ratio of individual resistance to total resistance is the same as the ratio of individual voltage drop to total supply voltage in a voltage divider circuit. This is known as the voltage divider formula , and it is a short-cut method for determining voltage drop in a series circuit without going through the current calculation(s) of Ohm’s Law.
Using this formula, we can re-analyze the example circuit’s voltage drops in fewer steps:
Voltage dividers find wide application in electric meter circuits, where specific combinations of series resistors are used to “divide” a voltage into precise proportions as part of a voltage measurement device.
One device frequently used as a voltage-dividing component is the potentiometer , which is a resistor with a movable element positioned by a manual knob or lever. The movable element, typically called a wiper , makes contact with a resistive strip of material (commonly called the slidewire if made of resistive metal wire) at any point selected by the manual control:
The wiper contact is the left-facing arrow symbol drawn in the middle of the vertical resistor element. As it is moved up, it contacts the resistive strip closer to terminal 1 and further away from terminal 2, lowering resistance to terminal 1 and raising resistance to terminal 2. As it is moved down, the opposite effect results. The resistance as measured between terminals 1 and 2 is constant for any wiper position.
Rotary vs. Linear Potentiometers
Shown here are internal illustrations of two potentiometer types, rotary and linear:
Some linear potentiometers are actuated by straight-line motion of a lever or slide button. Others, like the one depicted in the previous illustration, are actuated by a turn-screw for fine adjustment ability. The latter units are sometimes referred to as trimpots , because they work well for applications requiring a variable resistance to be “trimmed” to some precise value. It should be noted that not all linear potentiometers have the same terminal assignments as shown in this illustration. With some, the wiper terminal is in the middle, between the two end terminals.
The following photograph shows a real, rotary potentiometer with exposed wiper and slidewire for easy viewing. The shaft which moves the wiper has been turned almost fully clockwise so that the wiper is nearly touching the left terminal end of the slidewire:
Here is the same potentiometer with the wiper shaft moved almost to the full-counterclockwise position, so that the wiper is near the other extreme end of travel:
If a constant voltage is applied between the outer terminals (across the length of the slidewire), the wiper position will tap off a fraction of the applied voltage, measurable between the wiper contact and either of the other two terminals. The fractional value depends entirely on the physical position of the wiper:
The Importance of Potentiometer Application
Just like the fixed voltage divider, the potentiometer’s voltage division ratio is strictly a function of resistance and not of the magnitude of applied voltage. In other words, if the potentiometer knob or lever is moved to the 50 percent (exact center) position, the voltage dropped between wiper and either outside terminal would be exactly 1/2 of the applied voltage, no matter what that voltage happens to be, or what the end-to-end resistance of the potentiometer is. In other words, a potentiometer functions as a variable voltage divider where the voltage division ratio is set by wiper position.
This application of the potentiometer is a very useful means of obtaining a variable voltage from a fixed-voltage source such as a battery. If a circuit you’re building requires a certain amount of voltage that is less than the value of an available battery’s voltage, you may connect the outer terminals of a potentiometer across that battery and “dial-up” whatever voltage you need between the potentiometer wiper and one of the outer terminals for use in your circuit:
When used in this manner, the name potentiometer makes perfect sense: they meter (control) the potential (voltage) applied across them by creating a variable voltage-divider ratio. This use of the three-terminal potentiometer as a variable voltage divider is very popular in circuit design.
Shown here are several small potentiometers of the kind commonly used in consumer electronic equipment and by hobbyists and students in constructing circuits:
The smaller units on the very left and very right are designed to plug into a solderless breadboard or be soldered into a printed circuit board. The middle units are designed to be mounted on a flat panel with wires soldered to each of the three terminals.
Here are three more potentiometers, more specialized than the set just shown:
The large “Helipot” unit is a laboratory potentiometer designed for quick and easy connection to a circuit. The unit in the lower-left corner of the photograph is the same type of potentiometer, just without a case or 10-turn counting dial. Both of these potentiometers are precision units, using multi-turn helical-track resistance strips and wiper mechanisms for making small adjustments. The unit on the lower-right is a panel-mount potentiometer, designed for rough service in industrial applications.
Review
- Series circuits proportion, or divide , the total supply voltage among individual voltage drops, the proportions being strictly dependent upon resistances: E Rn = E Total (R n / R Total )
- A potentiometer is a variable-resistance component with three connection points, frequently used as an adjustable voltage divider.


4.2: Conditions under which Integrals Approximate Partition Functions
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The Boltzmann equation gives the equilibrium fraction of particles in the \(i^{th}\) energy level, \(\epsilon_i\), as
\[\frac{N^{\textrm{⦁}}_i}{N}=\frac{g_i}{z}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]
so the fraction of particles in energy levels less than \(\epsilon_n\) is
\[f\left(\epsilon_n\right)=z^{-1}\sum^{n-1}_{i=1}{g_i}\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right) \nonumber \]
where \(z=\sum^{\infty }_{i=1}{g_i}\mathrm{exp}\left({\epsilon_i}/{kT}\right)\). We can represent either of these sums as the area under a bar graph, where the height and width of each bar are \(g_i\mathrm{exp}\left({\epsilon_i}/{kT}\right)\) and unity, respectively. If \(g_i\) and \(\epsilon_i\) can be approximated as continuous functions, this area can be approximated as the area under the continuous function \(y\left(i\right)=g_i\mathrm{exp}\left({\epsilon_i}/{kT}\right)\). That is,
\[\sum^{n-1}_{i=1}g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)\approx \int^n_{i=0}{g_i\mathrm{exp}\left(\frac{-\epsilon_i}{kT}\right)}di \nonumber \]
To evaluate this integral, we must know how both \(g_i\) and \(\epsilon_i\) depend on the quantum number, \(i\).
Let us consider the case in which \(g_i=1\) and look at the constraints that the \(\epsilon_i\) must satisfy in order to make the integral a good approximation to the sum. The graphical description of this case is sketched in Figure 1. Since \(\epsilon_i>\epsilon_{i-1}>0\), we have
\[e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0 \nonumber \]
For the integral to be a good approximation, we must have
\[e^{-\epsilon_{i-1}/kT}\gg e^{-\epsilon_{i-1}/kT}-e^{-\epsilon_i/kT}>0, \nonumber \]
which means that
\[1\gg 1-e^{-\Delta \epsilon /kT}>0 \nonumber \]
where \(\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\). Now,
\[e^x\approx 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots \nonumber \]
so that the approximation will be good if
\[1\gg 1-\left(1-\frac{\Delta \epsilon }{kT}+\dots \right) \nonumber \] or \[1\gg \frac{\Delta \epsilon }{kT} \nonumber \] or \[kT\gg \Delta \epsilon \nonumber \]
We can be confident that the integral is a good approximation to the exact sum whenever there are many pairs of energy levels, \(\epsilon_i\) and \(\epsilon_{i-1}\), that satisfy the condition
\[\Delta \epsilon =\epsilon_i-\epsilon_{i-1}\ll kT. \nonumber \]
If there are many energy levels that satisfy \(\epsilon_i\ll kT\), there are necessarily many intervals, \(\Delta \epsilon\), that satisfy \(\Delta \epsilon \ll kT\). In short, if a large number of the energy levels of a system satisfy the criterion \(\epsilon \ll kT\), we can use integration to approximate the sums that appear in the Boltzmann equation. In Section 24.3 , we use this approach and the energy levels for a particle in a box to find the partition function for an ideal gas.


1.5.20: Chemical Potentials- Salt Solutions- Ion-Ion Interactions
For most dilute aqueous salt solutions (at ambient temperature and pressure), mean ionic activity coefficients \(\gamma_{\pm}\) are less than unity. Ion-ion interactions within a real solution lower chemical potentials below those of salts in the corresponding ideal solutions. A quantitative treatment of this stabilisation is enormously important. In fact for almost the whole of the 20th Century, scientists have offered theoretical bases for expressing \(\ln \left(\gamma_{\pm}\right)\) as a function of the composition of a salt solution, temperature, pressure and electrical permittivity of the solvent.
In effect we offer as much information as demanded by the theory (e.g. molality of salt, nature of salt, permittivity of solvent, ion sizes, temperature, pressure, .....). We set the apparently simple task - please calculate the mean activity coefficient of the salt in this solution.
Many models and treatments have been proposed [1]. Most models start by considering a reference j-ion in an aqueous salt solution. In order to calculate the electric potential at the \(j\)-ion arising from all other ions in solution, we need to know the distribution of these ions about the \(j\)-ion. Unfortunately this distribution is unknown and so we need a model for this distribution. Further activity coefficients also reflect the impact of ions on water-water interactions in aqueous salt solutions. [2]
Footnotes
[1]
- R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edition revised,1959.
- H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions ,Reinhold, New York,1950, 2nd. edn.,1950, chapter 2. The analysis presented by Harned and Owen anticipates application to irreversible processes; e.g. electrolytic conductance of salt solutions. We confine attention to equilibrium properties.
[2] H. S. Frank, Z. Phys. Chem, 1965, 228 ,364.


20.3: Perspectives on the Phylogenetic Tree
Skills to Develop
- Describe horizontal gene transfer
- Illustrate how prokaryotes and eukaryotes transfer genes horizontally
- Identify the web and ring models of phylogenetic relationships and describe how they differ from the original phylogenetic tree concept
The concepts of phylogenetic modeling are constantly changing. It is one of the most dynamic fields of study in all of biology. Over the last several decades, new research has challenged scientists’ ideas about how organisms are related. New models of these relationships have been proposed for consideration by the scientific community.
Many phylogenetic trees have been shown as models of the evolutionary relationship among species. Phylogenetic trees originated with Charles Darwin, who sketched the first phylogenetic tree in 1837 (Figure \(\PageIndex{1}\)a), which served as a pattern for subsequent studies for more than a century. The concept of a phylogenetic tree with a single trunk representing a common ancestor, with the branches representing the divergence of species from this ancestor, fits well with the structure of many common trees, such as the oak (Figure \(\PageIndex{1}\)b). However, evidence from modern DNA sequence analysis and newly developed computer algorithms has caused skepticism about the validity of the standard tree model in the scientific community.
Limitations to the Classic Model
Classical thinking about prokaryotic evolution, included in the classic tree model, is that species evolve clonally. That is, they produce offspring themselves with only random mutations causing the descent into the variety of modern-day and extinct species known to science. This view is somewhat complicated in eukaryotes that reproduce sexually, but the laws of Mendelian genetics explain the variation in offspring, again, to be a result of a mutation within the species. The concept of genes being transferred between unrelated species was not considered as a possibility until relatively recently. Horizontal gene transfer (HGT), also known as lateral gene transfer, is the transfer of genes between unrelated species. HGT has been shown to be an ever-present phenomenon, with many evolutionists postulating a major role for this process in evolution, thus complicating the simple tree model. Genes have been shown to be passed between species which are only distantly related using standard phylogeny, thus adding a layer of complexity to the understanding of phylogenetic relationships.
The various ways that HGT occurs in prokaryotes is important to understanding phylogenies. Although at present HGT is not viewed as important to eukaryotic evolution, HGT does occur in this domain as well. Finally, as an example of the ultimate gene transfer, theories of genome fusion between symbiotic or endosymbiotic organisms have been proposed to explain an event of great importance—the evolution of the first eukaryotic cell, without which humans could not have come into existence.
Horizontal Gene Transfer
Horizontal gene transfer (HGT) is the introduction of genetic material from one species to another species by mechanisms other than the vertical transmission from parent(s) to offspring. These transfers allow even distantly related species to share genes, influencing their phenotypes. It is thought that HGT is more prevalent in prokaryotes, but that only about 2% of the prokaryotic genome may be transferred by this process. Some researchers believe such estimates are premature: the actual importance of HGT to evolutionary processes must be viewed as a work in progress. As the phenomenon is investigated more thoroughly, it may be revealed to be more common. Many scientists believe that HGT and mutation appear to be (especially in prokaryotes) a significant source of genetic variation, which is the raw material for the process of natural selection. These transfers may occur between any two species that share an intimate relationship (Table \(\PageIndex{1}\)).
Table \(\PageIndex{1}\): Summary of Mechanisms of Prokaryotic and Eukaryotic HGT
| Mechanism | Mode of Transmission | Example | |
|---|---|---|---|
| Prokaryotes | transformation | DNA uptake | many prokaryotes |
| transduction | bacteriophage (virus) | bacteria | |
| conjugation | pilus | many prokaryotes | |
| gene transfer agents | phage-like particles | purple non-sulfur bacteria | |
| Eukaryotes | from food organisms | unknown | aphid |
| jumping genes | transposons | rice and millet plants | |
| epiphytes/parasites | unknown | yew tree fungi | |
| from viral infections |
HGT in Prokaryotes
The mechanism of HGT has been shown to be quite common in the prokaryotic domains of Bacteria and Archaea, significantly changing the way their evolution is viewed. The majority of evolutionary models, such as in the Endosymbiont Theory, propose that eukaryotes descended from multiple prokaryotes, which makes HGT all the more important to understanding the phylogenetic relationships of all extant and extinct species.
The fact that genes are transferred among common bacteria is well known to microbiology students. These gene transfers between species are the major mechanism whereby bacteria acquire resistance to antibiotics. Classically, this type of transfer has been thought to occur by three different mechanisms:
- Transformation: naked DNA is taken up by a bacteria
- Transduction: genes are transferred using a virus
- Conjugation: the use of a hollow tube called a pilus to transfer genes between organisms
More recently, a fourth mechanism of gene transfer between prokaryotes has been discovered. Small, virus-like particles called gene transfer agents (GTAs) transfer random genomic segments from one species of prokaryote to another. GTAs have been shown to be responsible for genetic changes, sometimes at a very high frequency compared to other evolutionary processes. The first GTA was characterized in 1974 using purple, non-sulfur bacteria. These GTAs, which are thought to be bacteriophages that lost the ability to reproduce on their own, carry random pieces of DNA from one organism to another. The ability of GTAs to act with high frequency has been demonstrated in controlled studies using marine bacteria. Gene transfer events in marine prokaryotes, either by GTAs or by viruses, have been estimated to be as high as 10 13 per year in the Mediterranean Sea alone. GTAs and viruses are thought to be efficient HGT vehicles with a major impact on prokaryotic evolution.
As a consequence of this modern DNA analysis, the idea that eukaryotes evolved directly from Archaea has fallen out of favor. While eukaryotes share many features that are absent in bacteria, such as the TATA box (found in the promoter region of many genes), the discovery that some eukaryotic genes were more homologous with bacterial DNA than Archaea DNA made this idea less tenable. Furthermore, the fusion of genomes from Archaea and Bacteria by endosymbiosis has been proposed as the ultimate event in eukaryotic evolution.
HGT in Eukaryotes
Although it is easy to see how prokaryotes exchange genetic material by HGT, it was initially thought that this process was absent in eukaryotes. After all, prokaryotes are but single cells exposed directly to their environment, whereas the sex cells of multicellular organisms are usually sequestered in protected parts of the body. It follows from this idea that the gene transfers between multicellular eukaryotes should be more difficult. Indeed, it is thought that this process is rarer in eukaryotes and has a much smaller evolutionary impact than in prokaryotes. In spite of this fact, HGT between distantly related organisms has been demonstrated in several eukaryotic species, and it is possible that more examples will be discovered in the future.
In plants, gene transfer has been observed in species that cannot cross-pollinate by normal means. Transposons or “jumping genes” have been shown to transfer between rice and millet plant species. Furthermore, fungal species feeding on yew trees, from which the anti-cancer drug TAXOL® is derived from the bark, have acquired the ability to make taxol themselves, a clear example of gene transfer.
In animals, a particularly interesting example of HGT occurs within the aphid species (Figure \(\PageIndex{2}\)). Aphids are insects that vary in color based on carotenoid content. Carotenoids are pigments made by a variety of plants, fungi, and microbes, and they serve a variety of functions in animals, who obtain these chemicals from their food. Humans require carotenoids to synthesize vitamin A, and we obtain them by eating orange fruits and vegetables: carrots, apricots, mangoes, and sweet potatoes. On the other hand, aphids have acquired the ability to make the carotenoids on their own. According to DNA analysis, this ability is due to the transfer of fungal genes into the insect by HGT, presumably as the insect consumed fungi for food. A carotenoid enzyme called a desaturase is responsible for the red coloration seen in certain aphids, and it has been further shown that when this gene is inactivated by mutation, the aphids revert back to their more common green color (Figure \(\PageIndex{2}\)).
Genome Fusion and the Evolution of Eukaryotes
Scientists believe the ultimate in HGT occurs through genome fusion between different species of prokaryotes when two symbiotic organisms become endosymbiotic. This occurs when one species is taken inside the cytoplasm of another species, which ultimately results in a genome consisting of genes from both the endosymbiont and the host. This mechanism is an aspect of the Endosymbiont Theory, which is accepted by a majority of biologists as the mechanism whereby eukaryotic cells obtained their mitochondria and chloroplasts. However, the role of endosymbiosis in the development of the nucleus is more controversial. Nuclear and mitochondrial DNA are thought to be of different (separate) evolutionary origin, with the mitochondrial DNA being derived from the circular genomes of bacteria that were engulfed by ancient prokaryotic cells. Mitochondrial DNA can be regarded as the smallest chromosome. Interestingly enough, mitochondrial DNA is inherited only from the mother. The mitochondrial DNA degrades in sperm when the sperm degrades in the fertilized egg or in other instances when the mitochondria located in the flagellum of the sperm fails to enter the egg.
Within the past decade, the process of genome fusion by endosymbiosis has been proposed by James Lake of the UCLA/NASA Astrobiology Institute to be responsible for the evolution of the first eukaryotic cells (Figure \(\PageIndex{3}\)a). Using DNA analysis and a new mathematical algorithm called conditioned reconstruction (CR), his laboratory proposed that eukaryotic cells developed from an endosymbiotic gene fusion between two species, one an Archaea and the other a Bacteria. As mentioned, some eukaryotic genes resemble those of Archaea, whereas others resemble those from Bacteria. An endosymbiotic fusion event, such as Lake has proposed, would clearly explain this observation. On the other hand, this work is new and the CR algorithm is relatively unsubstantiated, which causes many scientists to resist this hypothesis.
More recent work by Lake (Figure \(\PageIndex{3}\)b) proposes that gram-negative bacteria, which are unique within their domain in that they contain two lipid bilayer membranes, indeed resulted from an endosymbiotic fusion of archaeal and bacterial species. The double membrane would be a direct result of the endosymbiosis, with the endosymbiont picking up the second membrane from the host as it was internalized. This mechanism has also been used to explain the double membranes found in mitochondria and chloroplasts. Lake’s work is not without skepticism, and the ideas are still debated within the biological science community. In addition to Lake’s hypothesis, there are several other competing theories as to the origin of eukaryotes. How did the eukaryotic nucleus evolve? One theory is that the prokaryotic cells produced an additional membrane that surrounded the bacterial chromosome. Some bacteria have the DNA enclosed by two membranes; however, there is no evidence of a nucleolus or nuclear pores. Other proteobacteria also have membrane-bound chromosomes. If the eukaryotic nucleus evolved this way, we would expect one of the two types of prokaryotes to be more closely related to eukaryotes.
The nucleus-first hypothesis proposes that the nucleus evolved in prokaryotes first (Figure \(\PageIndex{4}\)a), followed by a later fusion of the new eukaryote with bacteria that became mitochondria. The mitochondria-first hypothesis proposes that mitochondria were first established in a prokaryotic host (Figure \(\PageIndex{4}\)b), which subsequently acquired a nucleus, by fusion or other mechanisms, to become the first eukaryotic cell. Most interestingly, the eukaryote-first hypothesis proposes that prokaryotes actually evolved from eukaryotes by losing genes and complexity (Figure \(\PageIndex{4}\)c). All of these hypotheses are testable. Only time and more experimentation will determine which hypothesis is best supported by data.
Web and Network Models
The recognition of the importance of HGT, especially in the evolution of prokaryotes, has caused some to propose abandoning the classic “tree of life” model. In 1999, W. Ford Doolittle proposed a phylogenetic model that resembles a web or a network more than a tree. The hypothesis is that eukaryotes evolved not from a single prokaryotic ancestor, but from a pool of many species that were sharing genes by HGT mechanisms. As shown in Figure \(\PageIndex{5}\)a, some individual prokaryotes were responsible for transferring the bacteria that caused mitochondrial development to the new eukaryotes, whereas other species transferred the bacteria that gave rise to chloroplasts. This model is often called the “ web of life .” In an effort to save the tree analogy, some have proposed using the Ficus tree (Figure \(\PageIndex{5}\)b) with its multiple trunks as a phylogenetic to represent a diminished evolutionary role for HGT.
Ring of Life Models
Others have proposed abandoning any tree-like model of phylogeny in favor of a ring structure, the so-called “ ring of life ” (Figure \(\PageIndex{6}\)); a phylogenetic model where all three domains of life evolved from a pool of primitive prokaryotes. Lake, again using the conditioned reconstruction algorithm, proposes a ring-like model in which species of all three domains—Archaea, Bacteria, and Eukarya—evolved from a single pool of gene-swapping prokaryotes. His laboratory proposes that this structure is the best fit for data from extensive DNA analyses performed in his laboratory, and that the ring model is the only one that adequately takes HGT and genomic fusion into account. However, other phylogeneticists remain highly skeptical of this model.
In summary, the “tree of life” model proposed by Darwin must be modified to include HGT. Does this mean abandoning the tree model completely? Even Lake argues that all attempts should be made to discover some modification of the tree model to allow it to accurately fit his data, and only the inability to do so will sway people toward his ring proposal.
This doesn’t mean a tree, web, or a ring will correlate completely to an accurate description of phylogenetic relationships of life. A consequence of the new thinking about phylogenetic models is the idea that Darwin’s original conception of the phylogenetic tree is too simple, but made sense based on what was known at the time. However, the search for a more useful model moves on: each model serving as hypotheses to be tested with the possibility of developing new models. This is how science advances. These models are used as visualizations to help construct hypothetical evolutionary relationships and understand the massive amount of data being analyzed.
Summary
The phylogenetic tree, first used by Darwin, is the classic “tree of life” model describing phylogenetic relationships among species, and the most common model used today. New ideas about HGT and genome fusion have caused some to suggest revising the model to resemble webs or rings.
Glossary
- eukaryote-first hypothesis
- proposal that prokaryotes evolved from eukaryotes
- gene transfer agent (GTA)
- bacteriophage-like particle that transfers random genomic segments from one species of prokaryote to another
- genome fusion
- fusion of two prokaryotic genomes, presumably by endosymbiosis
- horizontal gene transfer (HGT)
- (also, lateral gene transfer) transfer of genes between unrelated species
- mitochondria-first hypothesis
- proposal that prokaryotes acquired a mitochondrion first, followed by nuclear development
- nucleus-first hypothesis
- proposal that prokaryotes acquired a nucleus first, and then the mitochondrion
- ring of life
- phylogenetic model where all three domains of life evolved from a pool of primitive prokaryotes
- web of life
- phylogenetic model that attempts to incorporate the effects of horizontal gene transfer on evolution


9.2: Basic Concepts
Normal Flora and Microbiome
Microorganisms occur naturally and are present everywhere in our environment. Some microorganisms live on the skin, in the nasopharynx, and in the gastrointestinal tract, but don’t become an infection unless the host becomes susceptible. These microorganisms are called normal flora . Over the past several, it has been discovered that every human being carries their own individual suite of microorganisms in and on their body referred to as their microbiome . A person’s microbiome is acquired at birth and evolves over their lifetime. It is different across body sites and between individuals. A person’s gut microbiome has recently been found to impact their immune system. [1] , [2]
Pathogens
Microorganisms that cause disease are called pathogens . There are four common types of pathogens, including viruses, bacteria, fungi, and parasites.
 After a host (i.e., the person) becomes infected by a virus, the virus invades the body’s cells and uses the components of the cell to replicate and produce more viruses. After the virus replication cycle is complete, the new viruses are released into the body, causing damage or destruction of the host’s cells. [3]
Antiviral medications can be used to treat some viral infections. Antibiotics do not kill viruses and are ineffective as a treatment for viral infections. See Figure 9.1 [4] for an image of a virus.
Bacteria
Bacteria are microorganisms made of a single cell. They are very diverse, have a variety of shapes and features, and have the ability to live in any environment, including your body. However, not all bacteria cause infections. Those that cause infection are called pathogenic bacteria. See Figure 9.2 [5] for an image of a bacterium called Escherichia coli (E. coli).
A patient is susceptible to bacterial infections when their immune system is compromised by chronic diseases or certain types of medications. Antibiotics are used to treat bacterial infections. However, some strains of bacteria have become resistant to antibiotics, making them difficult to treat. For example, infections caused by methicillin-resistant Staphylococcus Aureus (MRSA) are resistant to many types of antibiotics and have the capability of producing severe and life-threatening infections. MRSA infections usually require IV antibiotics and may require treatment for long periods of time. [6]
Fungi
There are millions of different fungal species on Earth. Fungi can be found everywhere in the environment, including indoors, outdoors, and on human skin, but only about 300 species cause infection when they overgrow. Candida albicans is a type of fungus that can cause oral thrush and vaginal yeast infections, especially in susceptible patients or those taking antibiotics. [7] See Figure 9.3 [8] for an image of oral thrush.
 This structure can make them harder to kill. Some new strains of fungal infections are proving to be especially dangerous, such as Candida auris , which is difficult to diagnose and treat, and can cause outbreaks in health care facilities. [9]
Parasites
Parasites are organisms that behave like tiny animals, living in or on a host, and feeding at the expense of the host. Three main types of parasites can cause disease in humans. These include the following:
- Protozoa: Single-celled organisms that can live and multiply in your body
- Helminths: Multi-celled organisms that can live inside or outside your body and are commonly known as worms
- Ectoparasites: Multi-celled organisms that live on or feed off skin, including ticks and mosquitos
Parasites can be spread several ways, including through contaminated soil, water, food, and blood, as well as through sexual contact and insect bites. [10] See Figure 9.4 [11] for an image of a helminth infection causing intestinal obstruction in a child.
- Davis, C. P. Normal flora. (1996). In S. Baron (Ed.), Medical Microbiology (4th ed.). University of Texas Medical Branch at Galveston. https://www.ncbi.nlm.nih.gov/books/NBK7617/ ↵
- This work is a derivative of Microbiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/microbiology/pages/1-introduction ↵
- This work is a derivative of Microbiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/microbiology/pages/1-introduction ↵
- “ 3D_medical_animation_corona_virus.jpg ” by https://www.scientificanimations.com is licensed under CC BY-SA 4.0 ↵
- “ E._coli_Bacteria_(16578744517).jpg ” by NIAID is licensed under CC BY 2.0 ↵
- This work is a derivative of Microbiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/microbiology/pages/1-introduction ↵
- This work is a derivative of Microbiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/microbiology/pages/1-introduction ↵
- “ Human_tongue_infected_with_oral_candidiasis.jpg ” by James Heilman, MD is licensed under CC BY-SA 3.0 ↵
- Manoylov, M. K. (2020, November 6). What are cytokines? Live Science. https://www.livescience.com/what-are-cytokines.html ↵
- This work is a derivative of Microbiology by OpenStax and is licensed under CC BY 4.0 . Access for free at https://openstax.org/books/microbiology/pages/1-introduction ↵
- “ Piece_of_intestine,_blocked_by_worms_(16424898321).jpg ” by SuSanA Secretariat is licensed under CC BY 2.0 ↵


8.5: Equidistant Property
Recall that distance from a line \(\ell\) to a point \(P\) is defined as the distance from \(P\) to its foot point on \(\ell\).
Assume \(\triangle ABC\) is not degenerate. Then a point \(X\) lies on the bisector of external bisector of \(\angle ABC\) if and only if \(X\) is equidistant from the lines \((AB)\) and \((BC)\).
- Proof
-
We can assume that \(X\) does not lie on the union of \((AB)\) and \((BC)\). Otherwise the distance to one of the lines vanish; in this case \(X = B\) is the only point equidistant from the two lines.
Let \(Y\) and \(Z\) be the reflections of \(X\) across \((AB)\) and \((BC)\) respectively. Note that
\(Y \ne Z\).
Otherwise both lines \((AB)\) and \((BC)\) are perpendicular bisectors of \([XY]\). that is, \((AB) = (BC)\) which is impossible since \(\triangle ABC\) is not degeneate. By Proposition 5.4.1 ,
\(XB = YB = ZB\).
Note that \(X\) is equidistant from \((AB)\) and \((BC)\) if and only if \(XY = XZ\). Applying SSS and then SAS, we get that
\(\begin{array} {rcl} {XY} & = & {XZ.} \\ {} & \Updownarrow & {} \\ {\triangle BXY} & \cong & {\triangle BXZ.} \\ {} & \Updownarrow & {} \\ {\measuredangle XBY} & \equiv & {\pm \measuredangle BXZ.} \end{array}\)
Since \(Y \ne Z\), we get that \(\measuredangle XBY \ne \measuredangle BXZ\); therefore
\[\measuredangle XBY = -\measuredangle BXZ.\]
By Proposition 5.4.1 , \(A\) lies on the bisector of \(\angle XBY\) and \(B\) lies on the bisector of \(\angle XBZ\); that is,
\(2 \cdot \measuredangle XBA \equiv \measuredangle XBY\), \(2 \cdot \measuredangle XBC \equiv \measuredangle XBZ.\)
By 8.5.1,
\(2 \cdot \measuredangle XBA \equiv -2 \cdot \measuredangle XBC.\)
The last identity means either
\(\measuredangle XBA + \measuredangle XBC \equiv 0\) or \(\measuredangle XBA + \measuredangle XBC \equiv \pi,\)
and hence the result.


1.4: Frequency, Frequency Tables, and Levels of Measurement
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Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible.
Answers and Rounding Off
A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. For example, the average of the three quiz scores four, six, and nine is 6.3, rounded off to the nearest tenth, because the data are whole numbers. Most answers will be rounded off in this manner.
It is not necessary to reduce most fractions in this course. Especially in Probability Topics , the chapter on probability, it is more helpful to leave an answer as an unreduced fraction.
Levels of Measurement
The way a set of data is measured is called its level of measurement . Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):
- Nominal scale level
- Ordinal scale level
- Interval scale level
- Ratio scale level
Data that is measured using a nominal scale is qualitative . Categories, colors, names, labels and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful.
Smartphone companies are another example of nominal scale data. Some examples are Sony, Motorola, Nokia, Samsung and Apple. This is just a list and there is no agreed upon order. Some people may favor Apple but that is a matter of opinion. Nominal scale data cannot be used in calculations.
Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data.
Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are “excellent,” “good,” “satisfactory,” and “unsatisfactory.” These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations.
Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point.
Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like -10° F and -15° C exist and are colder than 0.
Interval level data can be used in calculations, but one type of comparison cannot be done. 80° C is not four times as hot as 20° C (nor is 80° F four times as hot as 20° F). There is no meaning to the ratio of 80 to 20 (or four to one).
Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded.
The data can be put in order from lowest to highest: 20, 68, 80, 92.
The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20.
Frequency
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:
5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.
Table lists the different data values in ascending order and their frequencies.
| DATA VALUE | FREQUENCY |
|---|---|
| 2 | 3 |
| 3 | 5 |
| 4 | 3 |
| 5 | 6 |
| 6 | 2 |
| 7 | 1 |
Definition: Relative Frequency
A frequency is the number of times a value of the data occurs. According to Table Table \(\PageIndex{1}\), there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.
Definition: Relative frequencies
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY |
|---|---|---|
| 2 | 3 | \(\frac{3}{20}\) or 0.15 |
| 3 | 5 | \(\frac{5}{20}\) or 0.25 |
| 4 | 3 | \(\frac{3}{20}\) or 0.15 |
| 5 | 6 | \(\frac{6}{20}\) or 0.30 |
| 6 | 2 | \(\frac{2}{20}\) or 0.10 |
| 7 | 1 | \(\frac{1}{20}\) or 0.05 |
The sum of the values in the relative frequency column of Table \(\PageIndex{2}\) is \(\frac{20}{20}\), or 1.
Definition: Cumulative Relative Frequency
Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table \(\PageIndex{3}\).
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 2 | 3 | \(\frac{3}{20}\) or 0.15 | 0.15 |
| 3 | 5 | \(\frac{5}{20}\) or 0.25 | 0.15 + 0.25 = 0.40 |
| 4 | 3 | \(\frac{3}{20}\) or 0.15 | 0.40 + 0.15 = 0.55 |
| 5 | 6 | \(\frac{6}{20}\) or 0.30 | 0.55 + 0.30 = 0.85 |
| 6 | 2 | \(\frac{2}{20}\) or 0.10 | 0.85 + 0.10 = 0.95 |
| 7 | 1 | \(\frac{1}{20}\) or 0.05 | 0.95 + 0.05 = 1.00 |
The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.
Table \(\PageIndex{4}\) represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
| HEIGHTS (INCHES) | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 59.95–61.95 | 5 | \(\frac{5}{100} = 0.05\) | \(0.05\) |
| 61.95–63.95 | 3 | \(\frac{3}{100} = 0.03\) | \(0.05 + 0.03 = 0.08\) |
| 63.95–65.95 | 15 | \(\frac{15}{100} = 0.15\) | \(0.08 + 0.15 = 0.23\) |
| 65.95–67.95 | 40 | \(\frac{40}{100} = 0.40\) | \(0.23 + 0.40 = 0.63\) |
| 67.95–69.95 | 17 | \(\frac{17}{100} = 0.17\) | \(0.63 + 0.17 = 0.80\) |
| 69.95–71.95 | 12 | \(\frac{12}{100} = 0.12\) | \(0.80 + 0.12 = 0.92\) |
| 71.95–73.95 | 7 | \(\frac{7}{100} = 0.07\) | \(0.92 + 0.07 = 0.99\) |
| 73.95–75.95 | 1 | \(\frac{1}{100} = 0.01\) | \(0.99 + 0.01 = 1.00\) |
| Total = 100 | Total = 1.00 |
The data in this table have been grouped into the following intervals:
- 61.95 to 63.95 inches
- 63.95 to 65.95 inches
- 65.95 to 67.95 inches
- 67.95 to 69.95 inches
- 69.95 to 71.95 inches
- 71.95 to 73.95 inches
- 73.95 to 75.95 inches
This example is used again in Descriptive Statistics , where the method used to compute the intervals will be explained.
In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.
Exercise \(\PageIndex{1}\)
- From the Table \(\PageIndex{4}\), find the percentage of heights that are less than 65.95 inches.
- Find the percentage of heights that fall between 61.95 and 65.95 inches.
Answer
- If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are \(5 + 3 + 15 = 23\) players whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then \(\frac{23}{100}\) or 23%. This percentage is the cumulative relative frequency entry in the third row.
- Add the relative frequencies in the second and third rows: \(0.03 + 0.15 = 0.18\) or 18%.
Exercise \(\PageIndex{2}\)
Table \(\PageIndex{5}\) shows the amount, in inches, of annual rainfall in a sample of towns.
- Find the percentage of rainfall that is less than 9.01 inches.
- Find the percentage of rainfall that is between 6.99 and 13.05 inches.
| Rainfall (Inches) | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 2.95–4.97 | 6 | \(\frac{6}{50} = 0.12\) | \(0.12\) |
| 4.97–6.99 | 7 | \(\frac{7}{50} = 0.14\) | \(0.12 + 0.14 = 0.26\) |
| 6.99–9.01 | 15 | \(\frac{15}{50} = 0.30\) | \(0.26 + 0.30 = 0.56\) |
| 9.01–11.03 | 8 | \(\frac{8}{50} = 0.16\) | \(0.56 + 0.16 = 0.72\) |
| 11.03–13.05 | 9 | \(\frac{9}{50} = 0.18\) | \(0.72 + 0.18 = 0.90\) |
| 13.05–15.07 | 5 | \(\frac{5}{50} = 0.10\) | \(0.90 + 0.10 = 1.00\) |
| Total = 50 | Total = 1.00 |
- Answer
-
- \(0.56\) or \(56%\)
- \(0.30 + 0.16 + 0.18 = 0.64\) or \(64%\)
Exercise \(\PageIndex{3}\)
Use the heights of the 100 male semiprofessional soccer players in Table \(\PageIndex{4}\). Fill in the blanks and check your answers.
- The percentage of heights that are from 67.95 to 71.95 inches is: ____.
- The percentage of heights that are from 67.95 to 73.95 inches is: ____.
- The percentage of heights that are more than 65.95 inches is: ____.
- The number of players in the sample who are between 61.95 and 71.95 inches tall is: ____.
- What kind of data are the heights?
- Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players.
Remember, you count frequencies . To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.
Answer
- 29%
- 36%
- 77%
- 87
- quantitative continuous
- get rosters from each team and choose a simple random sample from each
Exercise \(\PageIndex{4}\)
From Table \(\PageIndex{5}\), find the number of towns that have rainfall between 2.95 and 9.01 inches.
- Answer
-
\(6 + 7 + 15 = 28\) towns
Collaborative Exercise \(\PageIndex{7}\)
In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:
- What percentage of the students in your class have no siblings?
- What percentage of the students have from one to three siblings?
- What percentage of the students have fewer than three siblings?
Example \(\PageIndex{7}\)
Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table \(\PageIndex{6}\) was produced:
| DATA | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 3 | 3 | \(\frac{3}{19}\) | 0.1579 |
| 4 | 1 | \(\frac{1}{19}\) | 0.2105 |
| 5 | 3 | \(\frac{3}{19}\) | 0.1579 |
| 7 | 2 | \(\frac{2}{19}\) | 0.2632 |
| 10 | 3 | \(\frac{3}{19}\) | 0.4737 |
| 12 | 2 | \(\frac{2}{19}\) | 0.7895 |
| 13 | 1 | \(\frac{1}{19}\) | 0.8421 |
| 15 | 1 | \(\frac{1}{19}\) | 0.8948 |
| 18 | 1 | \(\frac{1}{19}\) | 0.9474 |
| 20 | 1 | \(\frac{1}{19}\) | 1.0000 |
- Is the table correct? If it is not correct, what is wrong?
- True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.
- What fraction of the people surveyed commute five or seven miles?
- What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?
Answer
- No. The frequency column sums to 18, not 19. Not all cumulative relative frequencies are correct.
- False. The frequency for three miles should be one; for two miles (left out), two. The cumulative relative frequency column should read: 0.1052, 0.1579, 0.2105, 0.3684, 0.4737, 0.6316, 0.7368, 0.7895, 0.8421, 0.9474, 1.0000.
- \(\frac{5}{19}\)
- \(\frac{7}{19}\), \(\frac{12}{19}\), \(\frac{7}{19}\)
Exercise \(\PageIndex{8}\)
Table \(\PageIndex{5}\) represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year?
- Answer
-
\(\frac{9}{50}\)
Example \(\PageIndex{9}\)
Table \(\PageIndex{7}\) contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.
| Year | Total Number of Deaths |
|---|---|
| 2000 | 231 |
| 2001 | 21,357 |
| 2002 | 11,685 |
| 2003 | 33,819 |
| 2004 | 228,802 |
| 2005 | 88,003 |
| 2006 | 6,605 |
| 2007 | 712 |
| 2008 | 88,011 |
| 2009 | 1,790 |
| 2010 | 320,120 |
| 2011 | 21,953 |
| 2012 | 768 |
| Total | 823,356 |
Answer the following questions.
- What is the frequency of deaths measured from 2006 through 2009?
- What percentage of deaths occurred after 2009?
- What is the relative frequency of deaths that occurred in 2003 or earlier?
- What is the percentage of deaths that occurred in 2004?
- What kind of data are the numbers of deaths?
- The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers?
Answer
- 97,118 (11.8%)
- 41.6%
- 67,092/823,356 or 0.081 or 8.1 %
- 27.8%
- Quantitative discrete
- Quantitative continuous
Exercise \(\PageIndex{10}\)
Table \(\PageIndex{8}\) contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.
| Year | Total Number of Crashes | Year | Total Number of Crashes |
|---|---|---|---|
| 1994 | 36,254 | 2004 | 38,444 |
| 1995 | 37,241 | 2005 | 39,252 |
| 1996 | 37,494 | 2006 | 38,648 |
| 1997 | 37,324 | 2007 | 37,435 |
| 1998 | 37,107 | 2008 | 34,172 |
| 1999 | 37,140 | 2009 | 30,862 |
| 2000 | 37,526 | 2010 | 30,296 |
| 2001 | 37,862 | 2011 | 29,757 |
| 2002 | 38,491 | Total | 653,782 |
| 2003 | 38,477 |
Answer the following questions.
- What is the frequency of deaths measured from 2000 through 2004?
- What percentage of deaths occurred after 2006?
- What is the relative frequency of deaths that occurred in 2000 or before?
- What is the percentage of deaths that occurred in 2011?
- What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.
Answer
- 190,800 (29.2%)
- 24.9%
- 260,086/653,782 or 39.8%
- 4.6%
- 75.1% of all fatal traffic crashes for the period from 1994 to 2011 happened from 1994 to 2006.
References
- “State & County QuickFacts,” U.S. Census Bureau. quickfacts.census.gov/qfd/download_data.html (accessed May 1, 2013).
- “State & County QuickFacts: Quick, easy access to facts about people, business, and geography,” U.S. Census Bureau. quickfacts.census.gov/qfd/index.html (accessed May 1, 2013).
- “Table 5: Direct hits by mainland United States Hurricanes (1851-2004),” National Hurricane Center, http://www.nhc.noaa.gov/gifs/table5.gif (accessed May 1, 2013).
- “Levels of Measurement,” infinity.cos.edu/faculty/wood...ata_Levels.htm (accessed May 1, 2013).
- Courtney Taylor, “Levels of Measurement,” about.com, http://statistics.about.com/od/Helpa...easurement.htm (accessed May 1, 2013).
- David Lane. “Levels of Measurement,” Connexions, http://cnx.org/content/m10809/latest/ (accessed May 1, 2013).
Review
Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round off your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth.
In addition to rounding your answers, you can measure your data using the following four levels of measurement.
- Nominal scale level: data that cannot be ordered nor can it be used in calculations
- Ordinal scale level: data that can be ordered; the differences cannot be measured
- Interval scale level: data with a definite ordering but no starting point; the differences can be measured, but there is no such thing as a ratio.
- Ratio scale level: data with a starting point that can be ordered; the differences have meaning and ratios can be calculated.
When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these.
Exercise \(\PageIndex{11}\)
What type of measure scale is being used? Nominal, ordinal, interval or ratio.
- High school soccer players classified by their athletic ability: Superior, Average, Above average
- Baking temperatures for various main dishes: 350, 400, 325, 250, 300
- The colors of crayons in a 24-crayon box
- Social security numbers
- Incomes measured in dollars
- A satisfaction survey of a social website by number: 1 = very satisfied, 2 = somewhat satisfied, 3 = not satisfied
- Political outlook: extreme left, left-of-center, right-of-center, extreme right
- Time of day on an analog watch
- The distance in miles to the closest grocery store
- The dates 1066, 1492, 1644, 1947, and 1944
- The heights of 21–65 year-old women
- Common letter grades: A, B, C, D, and F
Answer
- ordinal
- interval
- nominal
- nominal
- ratio
- ordinal
- nominal
- interval
- ratio
- interval
- ratio
- ordinal
Glossary
- Cumulative Relative Frequency
- The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.
- Frequency
- the number of times a value of the data occurs
- Relative Frequency
- the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes to the total number of outcomes